A102491 -id:A102491 - cp's OEIS frontend

A007091 Numbers in base 5.

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 200, 201, 202, 203, 204, 210, 211, 212, 213, 214, 220, 221, 222, 223, 224, 230

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 4.
Thus nonnegative integers in base 10 whose doubling by normal addition or multiplication requires no carry operation. (End)
It appears that this sequence corresponds to the numbers n for which twice the sum of digits of n is the sum of digits of 2*n. - Rémy Sigrist, Nov 22 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007092 (base 6), A007093 (base 7), A007094 (base 8), A007095 (base 9).

Cf. A037454, A037462, A102491.

A007091 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,5): return op(convert(l,base,10,10^nops(l))): end: seq(A007091(n),n=0..58); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 5]], {n, 0, 60}]

a(n)=if(n<1,0,if(n%5,a(n-1)+1,10*a(n/5)))

apply( A007091(n)=fromdigits(digits(n,5)), [0..66]) \\ M. F. Hasler, Nov 18 2019

from gmpy2 import digits
def A007091(n): return int(digits(n,5)) # Chai Wah Wu, Dec 26 2021

a(0)=0 a(n)=10*a(n/5) if n==0 (mod 5) a(n)=a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

a(n) = n + 1/2*Sum_{k >= 1} 10^k*floor(n/5^k). Cf. A037454, A037462 and A102491. - Peter Bala, Dec 01 2016

A102489 Take the decimal representation of n and read it as if it were written in hexadecimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 112

Offset: 0

Reinhard Zumkeller, Jan 12 2005

List of numbers in base-16 representation that can be written with decimal digits.
Early in the sequence there are blocks recurring as a(n) = a(n-10)+16, but this pattern starts to fail when we reach 160, 161, ... with hex-representations A0, A1, ... which cannot be written with decimal digits. - Rick L. Shepherd, Jun 08 2012
Binary Coded Decimal (BCD) codes, common in electronics, when interpreted as plain binary-coded integers. For example, number 39 is BCD coded in two nibbles as 0011 1001 which is the binary expansion of 57; hence, taking into account the offset, a(1+39) = 57. - Stanislav Sykora, Jun 09 2012
Integers that avoid letters in their hexadecimal expansion. - Eliora Ben-Gurion, Aug 28 2019

			10 in decimal is 16 in base 16, so a(10)=16.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Hexadecimal

Complement of A102490; A102487, A102491, A102493, A122840.

Cf. A090725 (the subsequence of primes).

import Data.Maybe (fromJust, mapMaybe)
a102489 n = a102489_list !! (n-1)
a102489_list = mapMaybe dhex [0..] where
   dhex 0                         = Just 0
   dhex x | d > 9 || y == Nothing = Nothing
          | otherwise             = Just $ 16 * fromJust y + d
          where (x', d) = divMod x 16; y = dhex x'
-- Reinhard Zumkeller, Jul 06 2012

o10:= n -> min(padic:-ordp(n,2),padic:-ordp(n,5)):
d:= [0,seq((2*16^o10(n)+3)/5, n=1..1000)]:
ListTools:-PartialSums(d); # Robert Israel, Aug 30 2015

Table[FromDigits[IntegerDigits[n], 16], {n, 0, 70}] (* Ivan Neretin, Aug 12 2015 *)

a(n) - a(n-1) = (2*16^A122840(n) + 3)/5. - Robert Israel, Aug 30 2015

Edited by N. J. A. Sloane, Feb 08 2014 (changed definition, moved old definition to comment, changed offset and b-file).

A102487 Numbers in base-12 representation that can be written with decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Jan 12 2005

Numbers that are only in this sequence or only in A039274 but not in both are n= 131, 142, 275, 286, 419, 430 etc: see A039558. [From R. J. Mathar, Aug 30 2008]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Duodecimal
Wikipedia, Duodecimal

Complement of A102488; A102489, A102491, A102493.

Cf. A033048 (subsequence).

import Data.List (unfoldr)
a102487 n = a102487_list !! (n-1)
a102487_list = filter (all (< 10) . unfoldr (\x ->
   if x == 0 then Nothing else Just $ swap $ divMod x 12)) [0..]
-- Reinhard Zumkeller, Apr 18 2011

fQ[n_] := Last@ Union@ IntegerDigits[n, 12] < 10; Select[ Range[0, 86], fQ] (* Robert G. Wilson v, Apr 17 2012 *)

{for(testn=0,87,
lgt=1;
for(i=1,1000,if(12^i > testn,lgt=i;break()));
atst=testn;pasr=1;
for(j=1,lgt,lasd=atst%12;
if(lasd<10,atst=(atst-lasd)/12,pasr=0;break()));
if(pasr==1,print1(testn,", ")))}
\\ Douglas Latimer, Apr 17 2012

A102487_list = [int(str(x), 12) for x in range(10**6)] # Chai Wah Wu, Apr 09 2016

A037454 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)3^i is the base 3 representation of n.

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 12, 13, 14, 36, 37, 38, 42, 43, 44, 48, 49, 50, 72, 73, 74, 78, 79, 80, 84, 85, 86, 216, 217, 218, 222, 223, 224, 228, 229, 230, 252, 253, 254, 258, 259, 260, 264, 265, 266, 288, 289, 290, 294, 295, 296, 300, 301, 302, 432, 433, 434, 438

Offset: 0

Clark Kimberling

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A037462, A007091, A102491.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 6
    end
r end; [a(n) for n in 0:57] |> println # Peter Luschny, Jan 03 2021

seq(n + (1/2)*add(6^k*floor(n/3^k), k = 1..floor(ln(n)/ln(3))), n = 1..100); # Peter Bala, Dec 01 2016

t = Table[FromDigits[RealDigits[n, 3], 6], {n, 0, 100}]
(* Clark Kimberling, Aug 03 2012 *)

From Peter Bala, Dec 01 2016: (Start)
a(n) = n + 1/2*Sum_{k >= 1} 6^k*floor(n/3^k). Cf. A037462, A007091 and A102491.
a(0) = 0; a(n) = 6*a(n/3) if n == 0 (mod 3) else a(n) = a(n-1) + 1. (End)

Offset changed to 0 by Clark Kimberling, Aug 03 2012

A102492 Numbers in base-20 representation that cannot be written with decimal digits.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 130, 131, 132, 133

Offset: 1

Reinhard Zumkeller, Jan 12 2005

			112 = 5*20^1 + 12*20^0 = '5C', therefore 112 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Vigesimal
Wikipedia, Vigesimal

Complement of A102491; A102488, A102490, A102494.

import Data.List (unfoldr)
a102492 n = a102492_list !! (n-1)
a102492_list = filter (any (> 9) . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 20)) [0..]
-- Reinhard Zumkeller, Jun 27 2013

Select[Range[150],Max[IntegerDigits[#,20]]>9&] (* Harvey P. Dale, Mar 27 2021 *)

A102493 Numbers in base-60 representation that can be written with decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 300, 301, 302, 303, 304, 305, 306, 307, 308

Offset: 1

Reinhard Zumkeller, Jan 12 2005

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Sexagesimal

Complement of A102494; A102487, A102489, A102491.

import Data.List (unfoldr)
a102493 n = a102493_list !! (n-1)
a102493_list = filter (all (<= 9) . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 60)) [0..]
-- Reinhard Zumkeller, Jun 27 2013

Table[Range[60n,60n+9],{n,0,6}]//Flatten (* Harvey P. Dale, Jul 06 2024 *)

A118761 Fixed points of permutations A118757, A118758, A118759 and A118760.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 119, 138, 157, 176, 195, 310, 339, 358, 377, 396, 559, 578, 597, 779, 798, 999, 3130

Offset: 1

Reinhard Zumkeller, May 01 2006

nonn

A118757(a(n)) = A118758(a(n)) = A118759(a(n)) = A118760(a(n)) = a(n);

a(n) = A024657(n-1) = A102491(n) for n<=50.

Cf. A118767.

A024657 n written in fractional base 10/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 221, 222, 223, 224, 225, 226, 227

Offset: 0

David W. Wilson

To represent a number in base b, if a digit exceeds b-1, subtract b and carry 1. In fractional base a/b, subtract a and carry b.

Also numbers which are written the same in base 20/2 as in base 10. The sequence consists of numbers which have digits in {0,2,4,6,8} except that the unit digit can be any from {0,1,2,3,4,5,6,7,8,9} - Henry Bottomley, Nov 17 2000

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to fractional bases.

Cf. A007091, A058185, A058186, A102491, A118761.

a[n_] := a[n] = If[n == 0, 0, 10 * a[2 * Floor[n/10]] + Mod[n, 10]]; Array[a, 50, 0] (* Amiram Eldar, Aug 02 2025 *)

a(n) = if(n == 0, 0, 10 * a(n\10 * 2) + n % 10); \\ Amiram Eldar, Aug 02 2025

a(n) = A118761(n+1) for n < 50. - Reinhard Zumkeller, May 01 2006

A037462 a(n) = Sum_{i = 0..m} d(i)8^i, where Sum_{i = 0..m} d(i)4^i is the base 4 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 64, 65, 66, 67, 72, 73, 74, 75, 80, 81, 82, 83, 88, 89, 90, 91, 128, 129, 130, 131, 136, 137, 138, 139, 144, 145, 146, 147, 152, 153, 154, 155, 192, 193, 194, 195, 200, 201, 202, 203, 208, 209, 210

Offset: 0

Clark Kimberling

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A037454, A007091, A102491.

seq(n + (1/2)*add(8^k*floor(n/4^k), k = 1..floor(ln(n)/ln(4))), n = 1..100); # Peter Bala, Dec 01 2016

Table[FromDigits[RealDigits[n, 4], 8], {n, 0, 100}]
(* Clark Kimberling, Aug 14 2012 *)

From Peter Bala, Dec 01 2016: (Start):
a(n) = n + 1/2*Sum_{k >= 1} 8^k*floor(n/4^k). Cf. A037454, A007091 and A102491.
a(0) = 0; a(n) = 8*a(n/4) if n == 0 (mod 4) else a(n) = a(n-1) + 1. (End)

Offset changed to 0 by Clark Kimberling, Aug 14 2012

cp's OEIS Frontend

A007091 Numbers in base 5.

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A102489 Take the decimal representation of n and read it as if it were written in hexadecimal.

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A102487 Numbers in base-12 representation that can be written with decimal digits.

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A037454 a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*3^i is the base 3 representation of n.

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A102492 Numbers in base-20 representation that cannot be written with decimal digits.

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A102493 Numbers in base-60 representation that can be written with decimal digits.

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A118761 Fixed points of permutations A118757, A118758, A118759 and A118760.

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A037454 a(n) = Sum_{i=0..m} d(i)6^i, where Sum_{i=0..m} d(i)3^i is the base 3 representation of n.

A037462 a(n) = Sum_{i = 0..m} d(i)8^i, where Sum_{i = 0..m} d(i)4^i is the base 4 representation of n.